The channel output QAM symbols \(Y=[Y_{\mathrm {c},1} \ldots Y_{\mathrm {c},n}]\)  are demapped into an L-value sequence \(L= [L_1 \ldots L_{(m_{\mathrm {s}}+m_{\mathrm {a}})n}]\)  by memoryless bit-metric decoding [1]},
where \(Y_{\mathrm {c},i}\in \mathbb {R}^2\)  and \(\mathbb {R}\)  denotes the real number set.
The receiver-side processing reverses the one on the transmitter side. QAM symbol demapping and FEC decoding are performed to recover the scrambled source sign bit sequence \(\hat{U}_{\mathrm {s}}\)  and the shaped amplitude bit sequence \(\hat{B}_{\mathrm {a}}\) . The recovered scrambled source sign bit sequence \(\hat{U}_{\mathrm {s}}\)  is bit-descrambled into the source sign bit sequence \(\hat{S}_{\mathrm {s}}\) . The shaping decoder, which is a hierarchical DM decoder here, converts \(\hat{B}_{\mathrm {a}}\)  into the flipped bit sequence \(\hat{F}\) . The bit-flipping is terminated based on the parity bit, i.e., if the parity bit is `1', all input bits are flipped and the parity bit is removed, otherwise the parity bit is just removed, to obtain the source amplitude bit sequence \(\hat{S}_{\mathrm {a}}\) . Finally, the source sign bit sequence \(\hat{S}_{\mathrm {s}}\)  and the source amplitude bit sequence \(\hat{S}_{\mathrm {a}}\)  are concatenated into the source information bit sequence \(\hat{S}\) .
